1. Field of the Invention
The present invention relates generally to frequency sensitivity analysis and optimum design for frequency robust resonators and particularly to microelectromechanical systems (MEMS) resonators.
2. Description of the Related Art
(Note: This application references a number of different publications as indicated throughout the specification by reference numbers enclosed in brackets, e.g., [x]. A list of these different publications ordered according to these reference numbers can be found below in Section 5 of the specification. Each of these publications is incorporated by reference herein.)
Resonators have been widely used as a key component in MEMS devices, such as in micro-gyroscopes [1,2,3], microvibromotors [4], micro-engines [5], and microwave systems [6]. Resonators are actuated, usually electrostatically, to oscillate at their natural resonant frequency. In many cases the robustness of the design frequency to process variations is one of the most important functional properties for the resonator design. Frequency accuracy of a resonator can directly affect the quality of the system in which it serves as a component. For the lateral vibrating rate gyroscopes, the frequency matching for their two vibrating modes is important for the output sensitivity. If frequency of any one of the modes shifts, the output signal""s accuracy will be decreased. Although symmetry in these gyroscopes helps the two modes to track to first order, it is useful to enhance the frequency matching by designing the resonant frequency to be insensitive to process variations.
In microvibromotors, several resonators impact a bar to make it move in the plane of the chip. If the impacting frequencies of the resonators are not harmonic, the motion of the bar will be unpredictable. Similarly, two orthogonal resonators actuate the previously mentioned micro-engine. The rotational stability of the engine is affected by the synchrony of these two resonators. Finally, in microwave systems, the resonator is used as an IF or RF filter. Therefore, the frequency accuracy of the resonator in this application is particularly important as its frequency determines the system performance in a fundamental way.
Several factors affect the stability of the resonator frequency, which include mass-loading [7], Brownian force [8] and fabrication variations [9]. The mass-loading that is the instantaneous differences in the rate of adsorption and desorption of contaminant molecules to and from the resonator surface causes mass fluctuation, which in turn leads to frequency fluctuation. Yong and Vig have given the mathematical expression to quantify the mass-loading effects on phase noise. A Brownian force is produced by the impingement of gas/air molecules on the structure, which Gabrielson has studied theoretically. Although it was known that the fabrication errors affect the frequency stability [11], fewer researchers have focused on this study because of the structure complexity and differences in micromachining methods.
With present micromachining techniques, the fabrication process variation in MEMS is inevitable and it will continue to be the case when devices are miniaturized to the point of process limitations. For example, the fabrication tolerance for the width of a typical suspension beam is reported to be about 10% in [10]. Even the same fabrication errors will cause different frequency variations for different resonator structures. There is a need in the art for design and fabrication processes that optimize the structure, tolerances and performance of MEMS resonators.
The present invention meets these needs.
The present invention comprises a frequency robust resonator including at least one beam having a first end, a second end, and a beam width and a proof mass having a proof mass area and a perimeter wherein the proof mass is affixed to the first end of the beam and the second end of the beam is affixed to a ground and the proof mass perimeter multiplied by the beam width is substantially equivalent to six times the proof mass area.
This invention applies to any resonant structure whose resonant frequency depends on a number of design parameters, one of which, is relatively uncertain. Let xcfx89(p0, p1, p2, . . . , pn) be a resonant frequency of a structure and p0, p1, p2, . . . , p be dimensional, material and other parameters. Without loss of generality, assume p0 is the parameter subject to manufacturing variations. The invention is to constrain the design parameters such that             ∂              ∂                  p          0                      ⁢          ω      ⁡              (                              p            0                    ,                      p            1                    ,                      p            2                    ,          …          ⁢                      xe2x80x83                    ,                      p            n                          )              =  0
so that the resonant frequency is insensitive, to first order, to manufacturing variations. Designs meeting this constraint are xe2x80x9cfrequency robustxe2x80x9d resonators. The invention applies to vibrating structures of all shapes and boundary conditions. For example, in addition to MEMS resonators, the invention applies to the design of bell, chimes, and other resonating structures.
In this invention, the frequency robustness for a folded-beam lateral vibrating resonator is analyzed. Based on the analysis, an optimum design method is presented for the resonator to obtain minimum frequency sensitivity. A simple relationship between the area and the perimeter of proof mass, and the beam width is derived for single material structures.